I actually wrote a long answer to this a while ago which should answer your question, which is a real toughie! The short answer is just that Western courts just didn’t have the right social context for imperial eunuchs to develop, which is kinda a lame answer I know. The right “mix” to get institutional eunuchs is a general situation where rulers are highly segregated, which creates a situation in which “liminal” people are needed to go between the sacred rulers and the outside world. The cheap answer is often that you need a harem to need eunuchs, but the Byzantines didn’t have imperial harems of women, so the anthropological theory of liminality is really the only thing that can begin to explain eunuchs cross-culturally.

There is also a general distaste for castrating your own children, so in many societies the eunuch tradition initially depends on there being an conquered or otherwise oppressed group of people that you can castrate to start it off. Most Western societies at the right time to get started with imperial eunuchs were not powerful, and eunuchs actually were a Western export product to the Near East for some time, this can be read about (rather briefly though!) in the monster Origins of the European Economy: Communications and Commerce AD 300–900 Over time in many societies the role of eunuch becomes ingrained enough (and with a decent enough social status) in society that they will stop importing eunuchs and start castrating their own children, usually however this is done by poorer people in society, so there’s still an a dynamic of oppressor/oppressed with eunuchs. The general poverty/low class origins of native eunuchs can be seen cross-culturally in many traditions, I’m sure you already know about the Chinese eunuchs being from poor areas.

I’m sorry to say Christian morals really had nothing to do with it. The Italian castrati were a Christian institution, and the slave traders that were making and exporting eunuch children from Africa to the (Muslim) Ottoman Empire were also Christian. The apostles and other early Christians were very accepting of eunuchs (and other slaves and oppressed people), such as with the story of the Ethiopian eunuch, which expressly shows that eunuchs were welcome to the apostles. Jesus also makes explicit reference to eunuchs. Christianity and eunuchs is one complicated relationship!

Sorry that this is kinda a long and rambly answer, you asked one of the million-dollar eunuch questions though!